The Missing Corners

Begin with an ordinary 8-by-8 chessboard, colored in the usual alternating pattern. Remove two squares: one corner and the corner diagonally opposite it. The remaining board has sixty-two squares. You also have thirty-one identical dominoes. Each domino covers exactly two squares that share a full edge; pieces may not overlap, extend beyond the board, or cover a missing corner. Since the number of domino squares equals the number of remaining board squares, the areas appear to match perfectly. Can the mutilated board be covered exactly by all thirty-one dominoes? Give a reason that proves your conclusion, rather than relying on a trial arrangement.